Modeling with Quadratic Equations Answers
A sporting goods store uses quadratic equations to monitor the daily cost and profit for various items it sells. The store’s daily profit, y, when soccer balls are sold at x dollars each, is modeled by . Why is there an interval over which the graph decreases?
The table shows some values that satisfy the quadratic equation. Which of the following is a true statement?
What is the vertex of the parabola? Round to the nearest hundredth.
The vertex is located at (8.33, 236.67). What does this point represent in context?
The vertex represents the maximum point of the quadratic function. In context, the x-coordinate (8.33) represents the price per soccer ball that will result in the maximum profit, and the y-coordinate (236.67) represents the maximum daily profit of $236.67.
What are the zeroes of the function? Round to the nearest hundredth.
What do the zeroes mean in context?
Suppose the store wants to earn a daily profit of $150 from the sale of soccer balls. To earn this profit, what price should the store charge for each soccer ball? Explain how to solve this problem.
To find the price that results in a daily profit of $150, set the profit equation equal to 150: 150 = -6x² + 100x - 180. Subtract 150 from both sides to get -6x² + 100x - 330 = 0. Using the quadratic formula, the solutions for x (the price) are approximately $4.53 and $12.13. Therefore, the store should charge either $4.53 or $12.13 per soccer ball to earn a $150 profit.
A sporting goods store uses quadratic equations to monitor the daily cost and profit for various items it sells. The store’s daily profit, y, when soccer balls are sold at x dollars each, is modeled by . Why is there an interval over which the graph decreases?
The table shows some values that satisfy the quadratic equation. Which of the following is a true statement?
What is the vertex of the parabola? Round to the nearest hundredth.
What are the zeroes of the function? Round to the nearest hundredth.
What do the zeroes mean in context?
Soccer balls go on sale for $7.50 each. The store also sells footballs and the manager wants to earn a daily profit of $400 from both items. The equation models the store’s daily profit, y, for selling footballs at x dollars. Explain how to find the price per football needed to meet the goal.
At $7.50 per soccer ball, the store earns $232.50 in daily profit. It needs another $400 − $232.50 = $167.50 from footballs. Set the football profit equation equal to $167.50 and solve 167.50 = −4x² + 80x − 150. The solutions are approximately $5.46 and $14.54 per football.
Which did you include in your response?
The quadratic equation y = –6x2 + 100x – 180 models the store’s daily profit, y, for selling soccer balls at x dollars.The quadratic equation y = –4x2 + 80x – 150 models the store’s daily profit, y, for selling footballs at x dollars. Use a graphing calculator to find the intersection point(s) of the graphs, and explain what they mean in the context of the problem.
The graphs intersect at approximately (1.84, −16.49) and (8.16, 236.49). At each intersection, the store earns the same profit from soccer balls and footballs at that price. For example, at about $8.16 per ball, the profit is about $236.49 for either type.
Which did you include in your response?
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